Loop assemblies having a central link

ABSTRACT

Improved reversibly expandable structures are formed from novel loop assemblies comprising a plurality of links, each of said links having at least one center pivot joint and a plurality of end pivot joints, each of at least two of said plurality of end pivot joints proximate to the outer edge of said loop assembly and connected to another link; each of said plurality of links being connected to another one of said plurality of links by at least two end pivot joints thereby forming a link pair, said loop assembly comprising at least three link pairs, each of said at least three link pairs connected to at least two other link pairs through at least one of said end pivot joints; each of said at least three link pairs connected to a central piece that is central to the loop assembly, said central piece being rotatable around a central axis, wherein the rotation of the central piece reversibly expands said loop assembly.

REFERENCE TO RELATED APPLICATION

[0001] This application claims the filing date of provisional patentapplication, Serial No. 60/267,240, filed Feb. 7, 2001.

BACKGROUND OF THE INVENTION

[0002] U.S. Pat. Nos. 4,942,700, 5,024,031, 6,082,056 and 6,219,974,hereby incorporated by reference as if fully disclosed herein, teachmethods for constructing reversibly expandable truss-structures in awide variety of shapes. The teachings therein have been used to buildstructures for diverse applications including architectural uses, publicexhibits and unique folding toys.

[0003] Utilizing the teachings of these patents, self-supportingstructures that maintain their overall shape as they expand or collapsein a synchronized manner may be constructed. A basic building block ofsuch structures is a “loop-assembly” which consists of three or morescissor units (disclosed in the '700 and '031 patents) or polygon-linkpairs (disclosed in the '056 and '974 patents), each consisting of apair of links that are pinned together at pivots lying near the middleof each link. Such a loop assembly comprises a ring of interconnectedlinks which can freely fold and unfold. However, at the center of such aring, a space or void is opened up as the ring expands, resulting inlessened structural stability.

[0004] It is, therefore, desirable to provide additional stability andstructural stability to such a loop assembly while retaining its abilityto expand and contract. It is also desirable to provide a centrallocation to provide a means to mechanically drive the entire assembly.

[0005] In accordance with the present invention a novel loop assembly ispresented that incorporates an additional useful feature. I havediscovered a way to provide a link-pair that lies at the center of theassembly. The middle pivot of this central link-pair is located at thecenter point of the assembly as a whole. Further, this pivot alwaysmaintains its location at the center of the loop assembly as it extendsand retracts.

[0006] Loop assemblies having such central link-pairs are betterstabilized and better self-supported than those without such a feature.The movement of structures built from such assemblies are bettersynchronized. Further, central link-pairs offer a conveniently placedpoint of control for folding structures. By simply introducing a rotarymotion of one such link relative to its paired link, a force istranslated in an even, symmetric fashion to the entire assembly, therebyopening and closing it. Thus, a motor may be conveniently attached toone central link and the motor shaft fixed to the paired central link toprovide a well-placed, stabilized means to drive the entire assembly.

[0007] Further, in addition to such mechanical improvements, suchcentral link-pairs lead to new functional applications, such as theconstruction of expanding wheels and spreading mechanisms.

SUMMARY OF THE INVENTION

[0008] In accordance with the present invention reversibly expandablestructures are formed from loop assemblies comprising interconnectedpairs of links which lie essentially on the surface of the structure orparallel to the plane of the surface of the structure. The links in theloop assembly have at least three pivot joints. At least some of thepolygon links however, have more than three pivot joints. One of thepivot joints on each link is a pivot joint for connecting to anotherlink to form a link pair. Each link also has at least one internal pivotjoint and one perimeter pivot joint. The internal pivot joints are usedfor interconnecting adjacent link pairs to form the loop assembly. Looplinks are additionally joined to a central piece located at the centerof the loop assembly. The central piece can be a circular constructionwith pivot points to which the pivots on the loop links are joined. Therotation of the central piece through a plurality of degrees clockwiseand counterclockwise, expands and contracts the entire loop assembly.The central piece can alternatively be constructed of scissor pairswhich open and close, resulting in the expansion and contracting of theloop assembly.

[0009] Loop assemblies can be joined together and/or to other link pairsthrough the perimeter pivot joints to form structures.

[0010] In one preferred embodiment of the present invention link pairsmay be connected to adjacent link pairs to form a loop assembly throughhub elements that are connected at the respective internal pivot jointsof the two link pairs. Similarly hubs elements can be used to connectloop assemblies together or loop assemblies to other link pairs throughthe perimeter pivot joints to form structures. In yet another embodimentof the present invention the pivot joints can be designed as livinghinges as described more fully below.

[0011] Structures built in accordance with the subject invention havespecific favorable properties, including: a) The ability to use highlyrigid materials rather than bending or distortion of the mechanicallinks, allowing for a smooth and fluid unfolding process; b) The use ofcompact, structurally favorable and inexpensive joints in the form ofsimple pivots; c) Retaining the strength and stability of the structureduring folding and unfolding since all movement in the structure is dueto the actual deployment process, without floppiness in the structure;d) A wide range of geometries; e) Inexpensive manufacture of structureswith flexible hinges that are formed continuously with the linksthemselves; f) Convenient assembly of structures of many differentshapes through kits of the necessary parts; g) The ability to create aspace-filling structure by arranging linkages in a three-dimensionalmatrix; h) Structures have additional stability and structural stabilitybecause of the central piece, while still retaining its ability toexpand and contract; and i) Structures have a central location toprovide a means to mechanically drive the entire assembly.

BRIEF DESCRIPTION OF THE DRAWINGS

[0012]FIG. 1 shows a link 1 having three pivots.

[0013]FIG. 2 shows link 1 joined to link 2 by pivot

[0014]FIG. 3 links 1 and 2 are shown rotated about their common pivot 3to a different relative position.

[0015]FIG. 4 links 1 and 2 are again shown in a different relativerotational position.

[0016]FIG. 5 shows a linkage consisting of four links which are joinedin a loop by pivots.

[0017]FIG. 6 shows the linkage of FIG. 5 in a different position.

[0018]FIG. 7 shows a linkage consisting of four links which are joinedin a loop by pivots.

[0019]FIG. 8 shows the linkage of FIG. 7 in a different position.

[0020]FIG. 9 shows a linkage consisting of six links joined in a looparrangement via pivots.

[0021]FIG. 10 shows the linkage of FIG. 9 in a different position.

[0022]FIG. 11 shows the linkage of FIG. 9 in yet another differentposition.

[0023]FIG. 12 shows a link having four pivots that form an isoscelestriangle and pivots that lie on a second triangle that is the mirrorimage of the first triangle formed.

[0024]FIG. 13 shows a linkage, which is an alternate embodiment of theinvention.

[0025]FIG. 14 shows the linkage of FIG. 13 drawn in a differentposition.

[0026]FIG. 15 shows a linkage consisting of five scissor pairs.

[0027]FIG. 16 shows the linkage of FIG. 15 in a different position.

[0028]FIGS. 17 and 18 show the linkage of FIG. 15 in perspective views.

[0029]FIG. 19 shows a linkage consisting of thirteen scissor pairs.

[0030]FIGS. 20 and 21 the linkage of FIG. 19 in two different positions.

[0031]FIGS. 22, 23 and 24 show perspective views of the linkage of FIG.19 in different positions.

[0032]FIG. 25 shows a scissor pair comprised of two links which have adifferent profile than those shown in the previous drawings.

[0033]FIG. 26 shows a linkage comprised of twelve perimeter scissorpairs and one central scissor pairs, all of whose pivot locations aresimilarly distributed to the linkage in FIG. 19.

[0034]FIG. 27 shows the linkage of FIG. 26 in a partially expandedposition.

[0035]FIG. 28 shows the linkage of FIG. 26 in a fully expanded position,an embodiment of the invention as an expanding wheel.

[0036]FIGS. 29, 30 and 31 each show a perspective view of the linkage ofFIG. 26.

[0037]FIG. 32 shows an alternate embodiment of the invention consistingof two scissor pairs which form a four bar linkage.

[0038]FIG. 33 shows a four sided linkage consisting of four perimeterscissor-pairs and one central scissor pair.

[0039]FIGS. 34 and 35 show the linkage of FIG. 33 in two differentpositions.

[0040]FIGS. 36, 37 and 38 show perspective views of the linkage of FIG.33 in different positions.

[0041]FIG. 39 shows an alternate embodiment of the invention consistingof two scissor pairs forming a four bar linkage.

[0042]FIG. 40 shows the linkage of FIG. 39 in a folded position.

[0043]FIG. 41 shows a four-sided linkage in a folded position.

[0044]FIG. 42 shows the linkage of FIG. 41 in an opened position.

[0045]FIGS. 43 and 44 show perspective views of the linkage of FIG. 41in two positions.

[0046]FIG. 45 shows an exploded view of the linkage of FIG. 41 in anopen position, with a motor shown ready to be attached to the centrallink.

[0047]FIG. 46 shows the linkage of FIG. 41 in its assembled form, wherethe motor has been joined to the central link and the shaft is fixed tothe central link.

[0048]FIG. 47 shows the linkage of FIG. 41 in its closed position, theshaft having been rotated and driving the entire linkage.

DETAILED DESCRIPTION OF THE INVENTION

[0049]FIG. 1 shows a link 1 having three pivots 3, 4 and 5. Linesconnecting these three pivots form an isosceles triangle 13, with pivots4 and 5 lying on the base and pivot 3 at the apex.

[0050]FIG. 2 shows link 1 joined to link 2 by pivot 3. Link 2 has anadditional pivot 6. Pivots 4, 5 and 6 are equidistant from pivot 3. Aline 7 is drawn through pivots 4 and 6. A second line 8 is drawn throughpivots 5 and 6.

[0051] In FIG. 3 links 1 and 2 are shown rotated about their commonpivot 3 to a different relative position. A line 9 is drawn throughpivots 4 and 6. A second line 10 is drawn through pivots 5 and 6. Theangle formed between Lines 9 and 10 is identical to the angle formedbetween lines 7 and 8 as shown in FIG. 1.

[0052] In FIG. 4 links 1 and 2 are again shown in a different relativerotational position. The similarly drawn lines 11 and 12 again form anangle, which is identical to that formed in FIG. 1 and FIG. 2.

[0053] In general, given two joined links, the first having three pivotswhose center points form an isosceles triangle, the second having twopivots whose distance is identical to the sides of that triangle, whichlinks are joined by the pivot at the apex of the first link, the angleformed by drawing lines that lie on the center points of the threeun-joined pivots is constant and unchanging for any relative anglebetween the two links.

[0054] In FIG. 5 a linkage 15 is shown consisting of four links 20, 22,24 and 26 which are joined in a loop by pivots 21, 23, 25 and 27respectively. The figure formed by connecting the center points of thesefour pivots is a parallelogram. Thus the linkage may be seen to be aparallel four-bar.

[0055] Link 20 has three pivots 27, 28 and 21 whose center points lie onthe vertices of isosceles triangles. Likewise link 22 has three pivots23, 29 and 21 which form an isosceles triangle, which is similar, but ofa different size, than that triangle formed by link 20. Line 30 is drawnthrough pivots 28 and 25. Line 31 is drawn through pivots 29 and 25.

[0056]FIG. 6 shows the linkage 15 in a different position. Lines 32 and33 are drawn through pivots 28, 25 and 29, 25 respectively. The angleformed between lines 30 and 31 shown in FIG. 5 is identical to the angleformed between lines 32 and 33 shown in FIG. 6.

[0057] In general, given a parallel four-bar linkage, each link beingjoined to two neighboring links, where two of the links have anadditional pivot each of which form an isosceles triangle with the othertwo pivots of that link, which two triangles thus formed are similar,the lines drawn between each of those additional pivots and the pivotconnecting the two links opposite forms an angle which is constant andunchanging for any relative position of the linkage.

[0058] In FIG. 7 a linkage 170 is shown consisting of four links 172,174, 176 and 178 which are joined in a loop by pivots 173, 175, 177 and179 respectively. The figure formed by connecting the center points ofthese four pivots is a rhomb. Thus the linkage may be seen to be aparallel four-bar with equal sides.

[0059] Link 172 has three pivots 179, 173 and 180 whose center pointslie on the vertices of isosceles triangles. Likewise link 174 has threepivots 175, 181 and 173 which form an isosceles triangle which is whosesides are the same length as that triangle formed by link 172, but whosebase is of different length. Line 185 is drawn through pivots 180 and177. Line 186 is drawn through pivots 181 and 177.

[0060]FIG. 8 shows the linkage 170 in a different position. Lines 187and 188 are drawn through pivots 180, 177 and 181, 177 respectively. Theangle formed between lines 187 and 188 shown in FIG. 7 is identical tothe angle formed between lines 186 and 185 shown in FIG. 8.

[0061] In general, given a equal-sided parallel four-bar linkage, eachlink being joined to two neighboring links, where two of the links havean additional pivot each of which form an isosceles triangle with theother two pivots of that link, which two triangles thus formed haveequal length sides, but bases of different lengths, the lines drawnbetween each of those additional pivots and the pivot connecting the twolinks opposite forms an angle which is constant and unchanging for anyrelative position of the linkage.

[0062] In FIG. 9 a linkage 38 is shown consisting of six links40,42,44,46,48 and 50 joined in a loop arrangement respectively viapivots 41,43,45,47,49 and 50. Link 40 may be seen to have three pivots:51, 41 and 55. Pivots 51 and 41 lie towards the perimeter of the loopassembly, while pivot 55 lies towards the interior of the loop assembly.Thus pivots of each of these types shall be hereinafter referred to asperimeter pivots and interior pivots respectively. Additionally tointerior pivot 55, there are five other interior pivots 56,57,58,59 and60.

[0063] Linkage 38 is further comprised of two centrally located links 64and 65. Three interior pivots 55,57 and 59 respectively connect links40,44 and 48 to central link 64. Three other interior pivots 56,58 and60 respectively connect links 42, 46 and 48 to central link 65. Centrallinks 64 and 65 are themselves attached by pivot 66.

[0064] Thus linkage 38 may be seen to consist of a region of outer linksand central links. The outer links have, in general, perimeter pivots,which serve to connect them into a loop arrangement, and interior pivotswhich server to connect the outer links to the central links. Thecentral links are pivotally attached to each other via a central pivot.

[0065]FIG. 10 shows linkage 38 in a different position; FIG. 11 showslinkage 38 in yet another different position. For each position of thelinkage, central links 64 and 65 can provide a convenient and stablepoint to drive the linkage; simply by rotating these two links relativeto each other, forces will be transmitted to the outer links in asymmetric fashion.

[0066]FIG. 12 shows a link 72 having four pivots 81, 82, 85 and 86.Pivots 81, 82 and 86 form an isosceles triangle and pivots 85, 82 and 86lie on a second triangle that is the mirror image of the first triangleformed.

[0067]FIG. 13 shows a linkage 70, which is an alternate embodiment ofthe invention. Linkage 70 comprised of four links 72, 74, 76 and 78,which are connected together by pivots 82, 87, 90 and 86 to form aparallel four-bar linkage. Like link 72, link 74 has four pivots 82, 87,84 and 85, which lie on the vertices of mirrored isosceles triangles.Line 91 is drawn between pivot 81 and 85. Line 92 is drawn between pivot84 and 83. The intersection of lines 91 and 92 is at the center point ofpivot 90.

[0068]FIG. 14 shows linkage 70 drawn in a different position. Line 93passes through pivots 81 and 85. Line 94 passes through pivots 84 and83. The intersection of lines 93 and 94 is again at the center point ofpivot 90, which lies opposite the two four-pivot links 72 and 74. Theangle formed between lines 93 and 94 is identical to that formed betweenline 91 and 92 in FIG. 13.

[0069] In general, given an equal-sided parallel four-bar linkage wheretwo of the links each have two pivots, and two other links each havefour pivots which lie on the vertices of two mirrored isoscelestriangles, and of those four pivots, two lie on the mirror line and areattached to neighboring links, and two are side pivots which remainunattached, two lines may be drawn each connecting between the sidepivots of neighboring links, which two lines will form an angle that isconstant and unchanging for any relative position of the linkage, andwill always intersect that pivot which lies opposite the two four-pivotlinks.

[0070]FIG. 15 shows a linkage 100 consisting of five scissor pairs 120,130, 140, 150 and 160. Each Scissor pair is comprised of two linksjoined by a centrally located pivot. For example, scissor pair 120 iscomprised of links 121 and 122 joined by pivot 103, the others aresimilarly formed.

[0071] Based on their general position and function, scissor-pairs 120,130, 140 and 150 shall be referred to as perimeter scissor-pairs,whereas 160 shall be referred to as a central scissor pair.

[0072] The links in scissor-pair 120 and 150 each have four pivots. Theyare joined both to their neighboring scissor-pairs—130 and 140respectively—and to the central scissor pair 160. They are thus calledcentrally attached perimeter pairs. The links in scissor pairs 130 and140 each have three pivots. They are joined only to their neighboringperimeter pairs, and are thus not centrally attached. A line drawnthrough side pivots 104 and 102 intersects central pivot 125. Likewise,lines drawn through 106,108 and 110,112 and 114,116 and 118,119respectively all intersect central pivot 125.

[0073]FIG. 16 shows linkage 100 in a different position. The five linesdrawn through side pivots 102,104 and 106,108 and 110,112 and 114,116and 118,119 respectively all intersect central pivot 125.

[0074]FIGS. 17 and 18 show linkage 100 in perspective views.

[0075]FIG. 19 shows a linkage 200 consisting of thirteen scissor pairs.There are twelve perimeter scissor-pairs forming a completeloop-assembly. Four perimeter pairs 205, 220, 235 and 250 are centrallyattached. Eight perimeter pairs 210, 215, 225, 230, 240,245, 255, 260are not centrally attached. The thirteenth scissor pair 265 is comprisedof two links 266 and 267, which are attached by center pivot 270. Pair265 is pivotally attached by a total of eight pivots to pairs205,220,235 and 250.

[0076]FIGS. 20 and 21 shows linkage 200 in two different positions. Byrotating links 266 and 267 relative to one another, central scissor pair265 may be seen to drive the entire assembly in a symmetric and stablefashion. Center pivot 270 remains at the geometric center of the entireassembly in all positions.

[0077]FIGS. 22, 23 and 24 show perspective views of linkage 200 indifferent positions.

[0078]FIG. 25 shows a scissor pair 302 comprised of two links 304 and306, which links have a different profile than those shown in theprevious drawings.

[0079]FIG. 26 shows a linkage 300 comprised of twelve perimeter scissorpairs and one central scissor pairs, all of whose pivot locations aresimilarly distributed to linkage 200. Due to the different profile ofthe individual links, the overall shape of the linkage is a circle.

[0080]FIG. 27 shows linkage 300 in a partially expanded position. FIG.28 shows linkage 300 in a fully expanded position. In this position theoverall shape of the linkage is a circle. Thus linkage 300 shows anembodiment of the invention as an expanding wheel.

[0081]FIGS. 29, 30 and 31 each show a perspective view of the linkage300.

[0082]FIG. 32 shows an alternate embodiment of the invention consistingof two scissor pairs 310 and 320 which form a four bar linkage 311.

[0083] In FIG. 33 a four sided linkage 370 is shown consisting of fourperimeter scissor-pairs 310, 330, 340 and 350 and one central scissorpair 360. FIGS. 34 and 35 show linkage 370 in two different positions.The center pivot of central scissor pair 360 always remains in thecenter of the linkage.

[0084]FIGS. 36, 37 and 38 show perspective views of linkage 370 indifferent positions.

[0085]FIG. 39 shows an alternate embodiment of the invention consistingof two scissor pairs 410 and 420 forming a four bar linkage 405. Therelative position of the pivots is identical to linkage 370 shown inFIG. 32, however links 412 and 414 each have an additional pivot,respectively 421 and 422. Lines 433, 431 and 432 drawn between 421, 422and 415, 417 and 416, 417 respectively may be seen to form a righttriangle.

[0086]FIG. 40 shows linkage 405 in a folded position. The triangleformed by lines passing through 421, 422 and 415, 417 and 416, 417respectively is similar to that formed in FIG. 39, but of differentsize.

[0087]FIG. 41 shows a four-sided linkage 500 in a folded position. FIG.42 shows linkage 500 in an opened position. Central scissor pair 505 maybe seen to drive linkage 500 by a relative rotation between each of itslinks.

[0088]FIGS. 43 and 44 show perspective views of linkage 500 in twopositions.

[0089]FIG. 45 shows an exploded view of linkage 500 in an open position,with motor 510 shown ready to be attached to central link 503. FIG. 46shows 500 in its assembled form, where motor 510 has been joined tocentral link 503 and the shaft 511 being fixed to central link 504.

[0090] In FIG. 47 linkage 500 is shown in its closed position, the shaft511 having been rotated and thereby driving the entire linkage.

[0091] It will be appreciated that the instant specification, drawingsand claims set forth by way of illustration and not limitation, and thatvarious modification and changes may be made without departing from thespirit and scope of the present invention.

What is claimed is:
 1. A reversibly expandable loop assembly comprising:a plurality of links, each of said links having at least one centerpivot joint and a plurality of end pivot joints, each of at least two ofsaid plurality of end pivot joints proximate to the outer edge of saidloop assembly and connected to another link; each of said plurality oflinks being connected to another one of said plurality of links by atleast two end pivot joints thereby forming a link pair, said loopassembly comprising at least three link pairs, each of said at leastthree link pairs connected to at least two other link pairs through atleast one of said end pivot joints; each of said at least three linkpairs connected to a central piece that is central to the loop assembly,said central piece being rotatable around a central axis, wherein therotation of the central piece reversibly expands said loop assembly. 2.A loop assembly according to claim 1, wherein said connection betweenadjacent link-pairs are direct pivot connections between said internalcorner pivot joints of said adjacent link-pairs.
 3. A loop assemblyaccording to claim 1, wherein said central piece comprises a scissorpair, wherein each half of said scissor pair is rotatably connected toeach other.
 4. A loop assembly according to claim 1, wherein the firstlink of said link pair having three pivots whose center points form anisosceles triangle, the second link of said link pair having two pivotswhose distance is identical to the sides of that triangle, which linksare joined by the pivot at the apex of the first link, the angle formedby drawing lines that lie on the center points of the three un-joinedpivots is constant and unchanging for any relative angle between thefirst and second links;
 5. A loop assembly according to claim 1, furthercomprising one or more hub elements for connecting said internal cornerpivot joints of said adjacent link pairs thereto.
 6. A structurereversibly expandable from a folded to an unfolded position withpartially unfolded positions therebetween, comprising at least two loopassemblies in accordance with claim 1, interconnected by said perimetercorner pivot joints.
 7. A structure according to claim 1, furthercomprising one or more link pairs having at least one perimeter pivotjoint, connected by said perimeter pivot joint to said loop assembly. 8.A structure according to claim 4 having a three dimensional shape.
 9. Astructure according to claim 6, further comprising a plurality of loopassemblies connected together by the corner joints of said links of saidloop assemblies, wherein at least one of said connection of saidplurality of loop assemblies joins together three or more of said links,to form a reversibly expandable three-dimensional matrix of links.
 10. Areversibly expandable three dimensional structure according to claim 6,wherein at least one of said loop assemblies in accordance with claim 1comprise at least two link pairs lying in different planes and whereinthe angle formed between any two planes of any two link pairs of saidloop assembly is substantially constant in any position of saidstructure.
 11. A loop assembly in accordance to claim 1, furthercomprising at least one sheet of material, said material comprising aplurality of stiff regions forming the links of the loop assembly andthin flexible regions forming corner pivot joints of said loop assemblyand said stiff regions connected together by one or more of said thinflexible regions.
 12. A loop assembly according to claim 9, furthercomprising two sheets of material, said two sheets of material joinedtogether by a plurality of center pivot joint connections.
 13. A loopassembly in accordance with claim 10, wherein said center pivot jointsare living hinges.
 14. A toy construction kit for building one or morereversibly expandable structures comprising a plurality of loopassemblies in accordance to claim 1, said loop assemblies includingperimeter corner joints for connecting any two of said plurality of loopassemblies, whereby said plurality of loop assemblies may be assembledtogether in various combinations forming reversibly expandablestructures of various shapes.
 15. A toy construction kit for buildingone or more reversibly expandable structures comprising a plurality oflinks for assembly into link pairs, each of said links having apolygonal profile with three or more corners, a center joint and atleast one corner pivot joint proximate to at least one of said cornersfor pivotally connecting two adjacent links, each of said link pairscapable of being connected to a central piece.
 16. A toy constructionkit according to claim 14, further comprising one or more hub elements,each of said hub elements to be shared by two or more of said links as apivotal connection therebetween.
 17. A toy construction kit according toclaim 14, further comprising at least two links having at least threecorner pivot joints.
 18. A toy construction kit according to claim 14,further comprising at least one motor to mechanically rotate at leastone central piece through a plurality of degrees clockwise and aplurality of degrees counter-clockwise.
 19. A loop assembly according toclaim 1, which further comprises an additional loop of connecting links,said connecting links each having an elongated profile, and each havingone center joint and two terminal joints, wherein: each terminal jointof each connecting link is pivotally connected to an adjacent connectinglink, thereby forming a loop; and each center joint of each connectinglink is pivotally connected to a polygon link in the loop assembly. 20.A reversibly expandable loop assembly comprising: a plurality of linkshaving a polygonal profile with three or more corners, a center pivotjoint and a plurality of corner pivot joints, each of at least two ofsaid plurality of corner pivot joints proximate to at least two of saidthree or more corners, respectively, said plurality of corner pivotjoints comprising at least one internal corner pivot joint and at leastone perimeter corner pivot joint proximately located to the outer edgeof said loop assembly; each of said plurality of links connected toanother one of said plurality of links at said center pivot jointthereby forming a link pair, said loop assembly comprising at leastthree link pairs; and each of said at least three link pairs connectedto at least two other link pairs, through at least one of said internalcorner pivot joints.
 21. A loop assembly according to claim 2, whereineach of said perimeter corner pivots has itself a pivotal connection toits corresponding polygon link pair, the axis of said pivotal connectionlying essentially within the plane of said polygon link pair.
 22. A loopassembly according to claim 20, wherein each perimeter corner pivot hasone or more flexible regions such that said flexible region of saidcorner pivot shall flexibly deform when brought together with theflexible region of a corner pivot belonging to another loop assembly,said flexible regions then snapping back to an unstrained condition asthe two corner pivots are brought into alignment, thus forming a pivotalattachment between said two perimeter corner pivots, thereby forming ahub element that acts essentially as a ball joint having threeintersecting axes of rotation.
 23. A toy construction kit for buildingreversibly expanding structures according to claim 12, comprising aplurality of loop assemblies according to claim
 20. 24. A toyconstruction kit for building reversibly expandable structures accordingto claim 14, further including at least one scissor-pair, said scissorpair comprised of two links pivotally connected together at a centerpivot joint, two ends of each of said two links each having a terminalpivot point that has itself a pivotal connection to its correspondinglink, the axis of said pivotal connection lying essentially within theplane formed by said scissor pair.
 25. A toy construction kit forbuilding reversibly expandable structures according to claim 14, furtherincluding means to attach two loop assemblies together in a stackedarrangement.
 26. A toy construction kit for building reversiblyexpandable structures according to claim 14, further including separatehub elements that may be attached to points on loop assemblies toprovide extra attachment points.
 27. A parallel four-bar linkageconsisting of four links, each link being pivotally connected to twoneighboring links, wherein at least two of the links have at least threepivots each, two of said three pivots are pivotally connected toneighboring links, with one additional pivot that may be unconnected,said three pivots of each link are located on the vertices of anisosceles triangle, wherein two lines may be drawn, each line connectingthe centerpoint of each of the additional pivots of the at least twolinks that have at least three pivots each, to that pivot which connectsthe two links that lie opposite the three-pivot links, and said twolines form an angle which is constant and unchanging for any relativeposition of the linkage.
 28. A parallel four-bar linkage consisting offour links, each link being joined to two neighboring links, where atleast two connected links have at least four pivots each, and the fourpivots of each link are located on the vertices of two mirroredisosceles triangles, whereby the two quadrilaterals thus formed aresimilar in profile, but not necessarily of the same size, two of thosefour pivots are connected to neighboring links which two pivots lie onthe symmetry line of each quadrilateral, with two additional side pivotsthat may be unconnected, whereby two lines may be drawn, each lineconnecting the centerpoints of two side pivots each belonging to adifferent four-pivot link, and said two lines form an angle which isconstant and unchanging for any relative position of the linkage.
 29. Alinkage according to claim
 28. where said two lines always intersectthat pivot which connects the two links that lie opposite the four-pivotlinks.
 30. A reversibly expandable loop assembly comprising: a pluralityof links having a polygonal profile with three or more corners, a centerpivot joint and a plurality of corner pivot joints, each of at least twoof said plurality of corner pivot joints proximate to at least two ofsaid three or more corners, respectively, said plurality of corner pivotjoints comprising at least one internal corner pivot joint and at leastone perimeter corner pivot joint proximately located to the outer edgeof said loop assembly; each of said plurality of links connected toanother one of said plurality of links at said center pivot jointthereby forming a link pair, said loop assembly comprising at leastthree link pairs: each of said at least three link pairs connected to atleast two link pairs, each of said at least three link pairs connectedto a central piece central to the loop assembly, each of said two linkpairs connected through at least one of said internal corner pivotjoints of a link in each of said two link pairs, said perimeter cornerpivot joint comprising a ball and socket arrangement.